The Economic Effects of Restrictions on Government Budget Deficits: Imperfect Private Credit Markets

Transcription

1 The Economic Effecs of Resricions on Governmen Budge Deficis: Imperfec Privae Credi Markes Chrisian Ghiglino Deparmen of Economics Queen Mary, Universiy of London Mile End Road, London E1 4NS UK Karl Shell Deparmen of Economics 402 Uris Hall Cornell Universiy Ihaca, NY USA Augus 27, 2001 Absrac Headnoe We consider a pure-exchange overlapping-generaions model wih many consumers per generaion many goods per period. As in Ghiglino Shell (2000), here is a governmen ha collecs axes, disribues ransfers faces budge defici resricions. We inroduce, for realism symmery wih he governmen, imperfecion in he privae credi markes. We find ha wih consrains on individual credi anonymous (i.e., non-personalized) lump-sum axes, srong (or global ) irrelevance of he governmen budge defici is no possible, weak irrelevance can hold only in very special siuaions. Wih credi consrains anonymous consumpion axes, weak irrelevance holds provided he number of ax insrumens is sufficienly large a leas one consumer s credi consrain is no binding. Journal of Economic Lieraure Classificaion Numbers: D51, E62, E52, H62, H63, O23. Keywords: Balanced Budge, Balanced-Budge Amendmen, Consumpion Taxes, Credi Consrains, Disorionary Taxes, Economic Policy, Governmen Budge Defici, Imperfec Credi Markes, Opimal Taxaion, Overlapping Generaions.

2 Corresponden: Chrisian Ghiglino, Deparmen of Economics, Queen Mary, Universiy of London, Mile End Road, London E1 4NS UK. elephone: ; fax: Running Head: Credi Raioning Defici Irrelevance 1

3 1. Inroducion summary In Ghiglino Shell (2000), we analyzed he economic effecs of consiuional or oher resricions on he governmen budge defici. We assumed ha privae agens have access o perfec markes for borrowing lending. This is a non-rivial assumpion. If governmen borrowing is resriced bu privae borrowing is unconsrained, hen he governmen can ease he effecs of is own borrowing resricions by in effec borrowing off he books by increasing he early-life axes on some individuals while a leas parially offseing his by increasing lae-life subsidies o he same individuals. In he real world, some consumers do face binding credi consrains or oher imperfecions in he borrowing marke. I is naural o ask how hese affec he governmen s abiliy o avoid resricions on is defici. In he presen paper, we assume ha he governmen individuals face credi resricions. The resricions on he governmen are from he consiuion or oher law, or from inernaional borrowing agreemens. The reasons for privae credi consrains include imperfec collaeral oher moral hazards. The sources of privae credi raioning will no be analyzed here. We simply assume ha here are exogenously given privae credi consrains which possibly differ across individuals. Following Ghiglino Shell (2000), we adop a pure-exchange overlapping-generaions model wih several consumers per generaion several commodiies per period. We allow for non-disoring lump-sum axes disoring consumpion axes. We also allow for he fac ha ax schedules canno be made perfecly individual-specific. In his paper, we focus for simpliciy on he perfecly anonymous case in which each consumer from he same generaion faces he same ax siuaion. We use wih apology he classic economic definiions 1 of relevance irrelevance applied in his case o governmen-budge-defici resricions. The governmen-budgedefici resricion is said o be irrelevan if he se of achievable allocaions is unaffeced by he resricion. Oherwise, he resricion is said o be relevan. Ofcourse, saying ha he resricion is irrelevan is no saying ha he resricion does no maer. If he resricion eiher direcly or indirecly affecs expecaions in such a manner ha i affecs he selecion of he equilibrium, hen he resricion does maer. If privae credi is unconsrained, here are lump-sum axes, hen budge defici resricions are (globally) irrelevan 2. If privae credi is consrained bu hese consrains are no binding on any individual he only ax insrumens are anonymous lump-sum axes, hen we have weak (local) irrelevance of he governmen budge resricions. If some credi consrains are binding, hen wih only anonymous lump-sum axes even local irrelevance is unlikely or impossible. If here are privae credi consrains, he case wih only consumpion axes is more ineresing. Surprisingly, consumpion axes, alhough disoring, are more likely o pro- 1 See Barro (1974). See also Ghiglino Shell (2000) he references herein. 2 See Ghiglino Shell (2000), Proposiion 5. 2

4 vide (a leas some form of) irrelevance han lump-sum axes. Wih credi consrains anonymous consumpion axes, here is weak (local) irrelevance if he number of ax insrumensissufficienly large a leas one consumer s credi consrain is no binding. This generalizes he resul for he case wih no privae credi consrains 3. In paricular, we show ha if he number of commodiies is no less han he sum of he number of individuals plus he number of individuals for whom credi raioning is binding, hen he defici resricion is weakly (locally) irrelevan 4. Why is weak (i.e. local) budge-defici irrelevance more likely wih anonymous consumpion axes han wih anonymous lump-sum axes? Their advanage o he governmen is ha hey can be used o ransfer income from one consumer o anoher in he same generaion, even when ax raes are anonymous. Such ransfers are impossible wih only anonymous lump-sum axaion. 2. The model We employ a pure-exchange overlapping-generaions model in which here are n differen consumers per generaion ` perishable commodiies per period. We suppose wihou loss of generaliy ha consumers live for wo periods. The governmen collecs axes, disribues ransfers (negaive axes), finances governmen consumpion. We focus on wo ypes of governmen insrumens: (non-disoring) lump-sum axes (disoring) consumpion axes. We assume ha lump-sum axes consumpion ax raes mus be he same for every member of a given generaion, bu ha consumpion axes can vary freely over he l commodiies. For he general case, see Ghiglino Shell (2000), which allows for more general consumer ax classes more general commodiy ax classes. We assume ha governmen consumpion of commodiies is exogenously deermined. I is denoed by he sequence g =(g 1,..., g,...)wihg R l + for =1, 2,... I is assumed ha use of capial markes is consrained, viz. each individual faces exogenously given consrains on his borrowing. Our se-up is based on he Samuelson (1958) overlapping-generaions model presened in Balasko Shell (1980, 1981, 1986), bu new ax insrumens he individual credi consrains mus be defined. As in Balasko Shell (1981), le m s h R be he lump-sum money ransfer o consumer h of generaion in period s; ifm s h is negaive, hen he consumer is paying a lump-sum ax. Following Ghiglino Shell (2000), we le τh si R be he presen (or nominal) ax rae levied on consumer h of generaion on his consumpion of commodiy i in period s. Le x s h =(xs1 h,..., xsi h,..., xsl h ) R`++ be he vecor of consumpion in period s by indi- 3 See Ghiglino Shell (2000), Proposiion If none of he privae credi consrains are binding, his inequaliy reduces (as i should) o he one in Proposiion 12 of Ghiglino Shell (2000) 3

5 vidual h of generaion ωh s =(ωs1 h,..., ωsi h,...,ws` h ) R`++ be he vecor of commodiy endowmens in period s of individual h from generaion for = 0, 1,..., s = 1, 2,..., h = 1,..., n. Le m s R be he money ransfer in period s o each consumer from generaion, τ s =(τ s1,...,τ si,...,τ sl ) R` be he vecor of anonymous consumpion ax raes in period s for consumers from generaion. Consumers from generaion 0 are alive in period 1, while consumers from generaion ( = 1, 2,..., ) are alive in periods + 1. Hence i is convenien o define he following vecors: x 0h = x 1 0 h R`++, ω 0 h = ω 1 0 h R`++, x h =(x h,x+1 h ) R2` ++, ω h =(ωh, ω+1 h ) R2 ++, ` m 0 = m 1 0 R, m =(m,m +1 ) R 2, τ 0 = τ 1 0 R`, τ =(τ, τ +1 ) R 2 `. Le p s =(p s1,..., p si,..., p s`) R`++ be he vecor of presen (before-ax) prices for commodiies available in period s le q s =(q s1,..., q si,..., q s` ) R`++ be he presen afer-ax vecor of commodiy prices facing consumers of generaion in period s. Define he afer-ax presen price vecors facing consumers of generaion =0, 1, 2, by q 0 = q 1 0 = p 1 + τ 0 R`++ for =0 (2.1) q = (q,q+h )=(p,p +1 )+(τ, τ +1 ) R++ 2` for =1, 2,... Then define he following quaniy price sequences: x =((x 0 h ) h=1 h=n,..., (x h) h=1 h=n,...), ω =((ω 0h ) h=n h=1,..., (ω h) h=n h=1,...), p =(p1,..., p,...), m =(m 0,..., m,...), τ =(τ 0,...,τ,...) q =(q 0,..., q,...). We assume ha he preferences of consumer h from generaion can be described by he uiliy funcion u h defined over he consumpion se of all sricly posiive x s (i.e. R`++ or R 2 ++) ` wih he properies: 4

6 (i) u h is wice differeniable wih sricly posiive firs-order derivaives wih corresponding negaive definie Hessian (ii) he closure of every indifference surface of u h is in he consumpion se (i.e. R`++ or R 2 ` ++). These raher sard assumpions simplify he comparaive saics 5. Noehawehave also assumed ha he endowmen of he consumer lies in his consumpion se, i.e. we have ω h is in R`++ or R 2` ++. Le b s h R + be he maximum credi available in money unis in period s o consumer h from generaion. The behavior of consumer h (h = 1, 2,..., n) fromgeneraion ( = 1, 2,...) is hen described by maximize u h (x h,x+1 h ) subjec o q h x h + xm h = p ω h + m h, +1,m q +1 h x +1 h + xh = p +1 ω +1 h + m +1 h, (2.2) x m h x m h b h, + x+1,m h = 0, where x sm h R is he gross money holding in period s by consumer h of generaion. The las equaion in (2.2) is he requiremen ha he consumer s indebedness be zero in his final period of life. The borrowing consrain is no binding on consumer h if in equilibrium x m h > b h. The inequaliy in (2.2) is he credi consrain. We have implicily assumed in wriing (2.2) ha he borrowing consrain of a leas one consumer is no binding, so ha we can use he usual no-arbirage argumen o esablish ha he presen price of money is consan, i.e., p,m = p +1,m = p m R + (2.3) 5 See Balasko (1988). See Balasko Shell (1980, 1981) for heir applicaion in overlappinggeneraions models. 5

7 where p s,m R + is he presen price of money in period s =1, 2,. Assuming ha he economy is in proper moneary equilibrium, we can wihou loss of generaliy se p m = 1. The nominal (coupon) rae of ineres on money is assumed wihou loss of generaliy o be zero. Hence he only reurn on holding money is he capial gain relaive o commodiies. Condiion (2.3) is hus ha money appreciae in value relaive o any commodiy a he commodiy rae of ineres. For consumers for which he credi resricion is no binding, Condiion (2.3) allows us o rewrie (2.2) somewha as Balasko Shell (1981) o yield maximize u h (x h,x+1 h ) subjec o q x h + q+1 x +1 h (2.4) = p ωh + p+1 ω +1 h + m + m +1 for h = 1, 2,..., n = 1, 2,..., where by choice of numeraire we se q01 1 =1. The ransfers m =(m,m +1 ) R 2 affec he behavior of he consumer only hrough he lifeime ransfer µ = m + m +1 R. I remains o describe he behavior of he older generaion ( = 0 )inperiod1. Consumer 0h maximizes his uiliy subjec o his one-period budge consrain: maximize u 0 h (x 1 0h ) subjec o q 1 0 x 1 0 h + x1m 0h = p1 ω 1 0 h + m1 0 x 1m 0h =0, (2.5) x 1 0h R` Fiscal policy We assume in his paper ha he governmen has a is disposal eiher anonymous lumpsum axaion or anonymous consumpion axaion. Thus, he governmen s fiscal policy is eiher he sequence of anonymous lump-sum ransfers m or he sequence of he consumpion ax raes τ. 6

8 Le d be he presen (also he dollar) value of he governmen budge defici in period. Hence we have for he case of lump-sum axaion d = p g + n m 1 + m for =1, 2,..., where n ishenumberofconsumerspergeneraion. consumpion axes Forhecaseof d = p g nx lx h=1 i=1 τ i 1 xi 1,h + τ i xi h for =1, 2,...Led denoe he sequence (d 1,..., d,...). Le δ be he presen ( nominal) value of he consiuionally imposed defici resricion (assumed for convenience in he form of an equaliy) in period. Le δ denoe he sequence (δ 1,..., δ,...). The budge defici resricion is hen 4. Equilibrium d = δ. We mainain hroughou his paper some srong assumpions. We suppose perfecforesigh on he par of consumers he governmen. We also suppose ha he governmen is able o perfecly commi o is announced fiscal policy. Nex we define equilibrium in he economy wih axes. Definiion Given he sequence of endowmens ω, he feasible fiscal policy m or τ, he exogenous consumpion g, he behavior of consumers described by he sysems (2.2), (2.4) (2.5), he numeraire choice yielding p 11 =1, he (furher) moneary normalizaion yielding p m = 1 he defici-resricion sequence δ, aconsiuional compeiive equilibrium is defined by a posiive price sequence p he allocaion sequence x such ha markes clear, i.e. we have Xh=n h=n g + (x 1,h + x,h )= X (ω 1,h + ω,h ) h=1 h=1 he defici resricion d = δ is saisfied. From Balasko Shell (1980), one migh expec ha he exisence of compeiive equilibrium o be guaraneed in nice overlapping-generaion models, bu his does no exend o our Definiion. There are hree reasons ha compeiive equilibrium as defined above could fail o exis. The firs reason is because we are seeking a proper moneary 7

9 equilibrium, one for which he price of money is sricly posiive. For a proper moneary equilibrium o exis he fiscal policy mus be bonafide 6. The second reason applies only o commodiy axaion. I migh no be possible o equilibrae supply dem while mainaining he posiiviy of he wo price sequences p q. The hird reason is ha equilibrium may fail o exis because of excessive governmen consumpion. When he model is saionary, i.e., preferences, endowmens, governmen consumpion are consan across generaions, one is emped o focus on equilibria in which allocaions are consan across periods. We provide separae definiions of he seady saeforhewoheaxregimes. Definiion L (Seady sae wih lump-sum axes): Le p =(p 1,,p, ) (R l ++) be he equilibrium sequence of commodiy prices when he fiscal policy is given by he sequence of lump-sum ransfers (m 1 0,m 1 1,m 2 1,,m,m +1, ). These describe a seady-sae equilibrium if here is a vecor p R l ++ a scalar β R ++ such ha p = β 1 p m = β 1 m 1 m +1 = β m 0 for =1, 2,, where m 1 = m 1 1 m 0 = m 1 0. Definiion C (Seady sae wih consumpion-axes): Le p =(p 1,,p, ) (R l +) be an equilibrium vecor of before-ax commodiy prices when he fiscal policy is given by he sequence of consumpion axes (τ0 1, τ1 1, τ1 2,, τ, τ +1, ) (R l ). These describe a seady-sae equilibrium if here is a vecor p R l ++ a scalar β R ++ such ha for =1, 2,,whereτ 1 = τ 1 1 τ 0 = τ 1 0. p = β 1 p τ = β 1 τ 1 τ +1 = β τ 0 When focusing on seady saes i makes sense o focus on budge deficis d = (d 1,,d, ) ha are consan in curren erms, so ha we have for a scalar d R d = β 1 d for =1, 2,... From Walras s law marke clearing, we have he wo seady-sae relaions: 6 See Balasko Shell (1981, 1986, 1993) Ghiglino Shell (2000). 8

10 " nx # (β 1) p x 1 h ωh 1 nm 1 + d =0 (SS-L) h=1 h=1 for lump-sum axaion, " nx (β 1) p # x 1 h lx ω1 h + τ 1i x 1i h + d = 0 (SS-C) for consumpion axaion, where τ 1i R is he ih componen of τ 1. The seady-sae condiions (SS-L) (SS-C) do no direcly involve governmen consumpion, bu seadysae g = g for =1, 2, is implied hrough he equilibrium allocaions prices. If d =0, from (SS-L) (SS-C), we have he familiar OG seady-sae resul ha eiher heineresraeiszero(β = 1) or aggregae savings is zero. Our aim is o find condiions under which he governmen is able o avoid he resricions on is defici wih changing neiher is own consumpion nor he consumpion of any privae consumer. When his is possible he defici resricion is said o be irrelevan. We recall he formal definiions given in Ghiglino Shell (2000). Definiion. Irrelevance of he defici resricion. Le g be governmen consumpions le x be an allocaion ha can be implemened as a compeiive equilibrium wih some feasible fiscal policy m (resp. τ) wih he resuling deficis given by he sequence d. The defici resricion d = δ is said o be irrelevan if for any oher defici resricion sequence δ 0 here exiss a feasible fiscal policy m (resp. τ) haimplemenshe allocaion x as a compeiive equilibrium is compaible wih g, bu wih he resuling defici given by he sequence δ 0. The above noion of irrelevance is very srong because i involves any possible defici sequence oher han he pre-reform, or baseline, defici d. In many siuaions such an irrelevance fails o obain. We pu forward a weaker noion of irrelevance. The firs characerisic of he weaker defici resricion is ha i only applies o a finie number of periods. Define δ(t )= (δ 1, δ 2,...,δ,...,δ T ) R T as a defici resricion of (finie) lengh T. For a compeiive equilibrium o be weakly consiuionally feasible he defici in period, d,musbeequal o δ if =1, 2,...,T, bu for >T, he defici is unresriced. The second characerisic of he weaker defici resricion is ha only resricions near o he base-line defici are considered, i.e., only period-by-period deficis ha are no oo differen from he baseline deficis are considered. In oher words, only a neighborhood of he original sequence is considered (in any opology, since he sequence is finie). According o he weaker noion of irrelevance only resricions of finie lengh δ(t )habelongoafirs T-period neighborhood of he base defici vecor d =(d 1,d 2,...,d,...,d T,...), denoed D T (d), are considered. Definiion. Weak irrelevance of he defici resricion Le g be he governmen consumpions le x be an allocaion ha can be implemened as a compeiive 9 i=1

11 equilibrium wih some feasible fiscal policy m (resp. τ) wih he resuling deficis given by he sequence d. Adefici resricion is said o be weakly irrelevan if for any T here is a se D T (d) such ha for all δ D T (d) here is a fiscal policy m 0 (resp. τ 0 ) ha implemens he allocaions x g, bu wih he resuling defici given by he sequence δ. Noe ha he ime horizon of he defici specificaion is arbirary may be any finie number T. 5. Relevance of governmen budge defici resricions wih lumpsum axes The relevance of governmen defici resricions is firs invesigaed in economies in which some consumers face credi consrains only lump-sum axaion is available. I is shown ha defici resricions are likely o be relevan unless he governmen can use non-anonymous axes. In he leading example considered below, he governmen uses only anonymous lump-sum axes ransfers hus has no way o rea differenly he consumers, so ha he likely oucome is ha some consumers are hur or benefied by he fiscal scheme. For general overlapping-generaions economies wih perfec borrowing markes lump-sum axes ransfers, resricions on he governmen budge have no impac on he se of equilibrium allocaions (see Ghiglino Shell (2000), Proposiion 5). The reason for his is ha in hese economies only he presen value of axes ransfers, no heir iming, maers o consumers. In his case, he governmen can borrow off he books from axpayers by adjusing he iming of individual axes ransfers. When credi resricions are included, weak (or local) irrelevance of he budge defici resricion obains if non-anonymous axes can be personalized o some consumer whose consrain is no binding. Being able o personalize axes is no always possible. If his is no possible, hen maers dramaically change. This is illusraed in he following example. For simpliciy, a saionary equilibrium is considered. This amouns o ignoring he ransiion pah. In oher words, in his example we will assume ha a suiable money ransfer is made so ha he economy sars a he seady sae only defici specificaions from = 2 onward are considered 7. Example (Relevance of governmen defici resricions when consumer credi is consrained): Le he economy be saionary wih one commodiy per period wo consumers. Perfecly anonymous lump-sum axaion is available. No oher ax insrumens are available. The wo consumers, 1 2, have log-linear uiliy funcions 7 Anoher, equivalen, way o view seady saes is o consider a model wih no beginning as well as no end (see Ghiglino Tvede (1995)). 10

12 u h = 1 /2 log x h + 1 /2 log x+1 h for h =1,2 = 1, 2,.... Endowmens are given by ω 1 = (ω 1, ω+1 1 )=(1, 20 ), ω 2 = (ω 2, ω +1 2 )=(0.75, 1 ). Consider generaion. Suppose ha he borrowing of consumer 1 is unconsrained, b 1 =, bu ha consumer 2 canno borrow, b 2 = 0. A a seady sae he individual dems of he consumers depend on he ineres facor β. When he credi consrain is no binding, he dems mus saisfy x,1 = 1+20β, 2 1 = 1+20β, 2 β x +1 x,2 = β, 2 x +1,2 = β. 2 β For β 0.75, we have x,2,x+1,2 =(0.75, 1) because hen he credi resricion is binding. Suppose ha g = 1 for =1, 2,... Wihou credi resricions β = β = solve he equilibrium equaions, bu β = is no an equilibrium ineres facor because he borrowing consrain for consumer 2 is violaed. The seady-sae equilibrium ineres facor is β = he corresponding equilibrium allocaions are x 1 = (x 1,x +1 1 )=(9.4408, ) 11

13 x 2 = (x 2,x+1 2 )=(0.75, 1 ). Since he governmen is no axing any consumer, he associaed defici is d = p g = p =( ) 1 in presen or nominal unis. Suppose now ha he governmen is required o balance is budge in every period, so ha d = δ = 0 for = 2, 3,.... We will show ha he new resricion on he deficis leads o a modificaion of he exising allocaion. Firs, noe ha in order o keep unchanged he consumpion of consumer 1, β should be unchanged a β = Now, he governmen can eiher ax he young or ax he old. Suppose firs ha consumers are axed in heir youh receive ransfers in heir old age. The procedure is similar o ha used in he proof of Proposiion 5 in Ghiglino Shell (2000). Since he consumers of he same generaion are perfecly anonymous for ax purposes, suppose ha we ax each youngequallywihalump-sumax m 2 2 > 0 no ax on he consumers born in he firs period, m 2 1 = 0. The governmen budge consrain is hen β 1 g +2m 2 2 =0soham 2 21 = m 2 22 = β/2 (or1/2 in curren erms) in order ha he defici be zero, d 2 =0. Noe in he nex period, hese same consumers have o be compensaed by a posiive ransfer of β/2 in presen erms, or 1/(2β) in curren erms. Afer he ransfer, he endowmens (in curren erms) of consumer 2 are ( , 1+0.5β). A β = , consumer 2 would sill like o borrow. However, due o he borrowing consrain his firs period consumpion is now The equilibrium allocaion has been affeced by he fiscal policy. The oher possibiliy is o subsidize he young ax he old. A similar reasoning shows ha also in his case he fiscal policy affecs he equilibrium allocaion. Therefore, he defici sequence is relevan. The previous example shows ha when some consumers are credi-consrained, anonymous lump-sum axes are no powerful enough o achieve irrelevance of he governmen budge defici. This is generalized in he following Proposiion (Relevance of he governmen defici resricions wih credi consrains): Le he allocaion x be implemened as a consiuional compeiive equilibrium wih a fiscal policy consising only of lump-sum axes ransfers compaible wih he defici resricion δ. If a leas one consumer s credi consrain is binding hen he defici sequence δ is weakly ( srongly) relevan. Oherwise, i is weakly irrelevan. Proof: If no consumer s credi consrain is binding, hen Proposiion 5 in Ghiglino Shell (2000) applies. However, in general as he governmen only employs anonymous axes any ransfer changes his acual borrowings (or savings) herefore affecs his dem for commodiies. Indeed, assume here are wo consumers ha h = 2 is he consumer whose credi consrain is binding (if he credi consrain of consumer 1 is also binding hen he defici resricion is obviously relevan). Consider consumer 2 firs. 12

14 His dems are he soluions o he problem maximize u 2 (x 2h,x+1 2h ) subjec o p x 2h + xm 2 = p ω 2 + m, +1,m p +1 x x2 = p +1 ω m +1, (5.1) x m x m 2 = b 2, +1,m 2 + x2 = 0. Wriing down he firs-order condiions, i is clear ha he allocaion is unchanged provided he normalized prices normalized firs second period incomes are unchanged by he new fiscal policy. This implies ha boh p /p 1 p +1 /p +1,1 are unaffeced by he policy change. The similar condiion for he incomes implies ha b m 2 + m = p,1 W b m 2 + m +1 = p,1 W 0 +1 where W,W 0 +1 Rl are unaffeced by he policy change. On he oher h, he governmen budge defici is Then, we obain d = m + m 1 + p,1 g δ = d = p,1 W + b 2 + p,1 W 0 1 b p,1 g which yields δ = p,1 (W + W 1 0 )+b 2 b p,1 g However, in order o keep consumer 1 unaffeced by he fiscal policy, p +1,1 /p,1 should also be kep consan, as p 1,1 =1, he enire sequence of prices should remain unchanged. As a resul, he defici sequence δ is enirely predeermined. Remark: If he governmen were able o use personalized lump-sum axes, hen he defici sequence δ would be weakly irrelevan. Indeed, renumber he consumers so ha consumer 1 is he unconsrained consumer reproduce he proof of Proposiion 5 in Ghiglino Shell (2000). 13

15 6. Resoring irrelevance wih consumpion axes In his secion we assume ha only anonymous axes on consumpion are available. The quesion is hen wheher he governmen is able o avoid he defici resricion wih hese insrumens even hough some consumers are credi consrained. As in he case wih unconsrained borrowing lending, he answer depends on he number of ax insrumens compared o he number of goals (consumers) on he duraion (in periods) of he resricion. We sar our analysis by an example. Example (Irrelevance of defici resricions in an economy wih several ax insrumens ): Consider a saionary, overlapping-generaions economy wih four commodiies per period (` = 4 ) wo consumers per generaion (n = 2 ). Assume ha he second consumer faces credi resricions while he oher has free access o he credi marke. The governmen has a consan consumpion in he firs good only, g = (g 1,g 2,g 3,g 4 )=(3, 0, 0, 0). Preferences endowmens of consumer h are given by: u h (x h,x +1 h ) = α kh log x k h + β kh log x +1k h ω h = (ω 0k h, ω1k h )4 wih ω1 01 = 300, ω = ω13 2 = 200, ω14 2 = 230, ω12 2 =120, ω01 ω1 12 = 500, ω2 02 = 1000, ωjk i = 100 for all oher h, j, k (α kh ),...,4 h=1,2, = 1/8 5/8 1/8 1/8 1/8 4/7 1/7 9/56 Wih anonymous consumpion axes we have τ k 1,1 = τ k 1,2 = τ k 1, (β kh ),...,4 h=1,2 = 2 =250, 1/4 1/4 1/5 6/20 1/5 1/4 1/4 6/20 τ k 1 = τ k 2 = τ k for k =1,.., 4 =1, 2,... For convenience, we look a seady-sae compeiive equilibrium. As noed earlier, we will only consider he periods from = 2 onward. Firs, we assume ha he governmen finances is consumpion by running a defici, i.e. we look a a seady sae wih τ k =0 τ 1 k =0. By resricing our aenion o prices of he form p k =(β) 1 p k,,..., 4, i can be shown ha he following se of allocaions prices represens a seady sae:. p 2 = , p 3 = , p 4 = , β =

16 (x 0k 1 )4 = ( , , , ) (x 1k 1 )4 = ( , , , ) (x 0k 2 ) 4 = ( , , , ) (x 1k 2 )4 = ( , , , ). The associaed defici is p 1 =(β) 1 p 1 =(β) 1 =( ) 1 in presen nominal erms. In curren unis, he savings are for consumer 1, for consumer 2 producing an aggregae savings of The issue is wheher (τ 1, τ ) 4 can be used in order o mee he defici requiremen δ =0inperiod wihou disurbing he allocaions given previously. Such a ax scheme mus a leas saisfy for each ( =2, 3,...) he following equaions x k 1,1 = (1 α 1 ) β k1w 11 p k + τ k 1 = x 0k 1, where x k 1 = α 1 α k1 W 1 p k + τ k = x 1k 1, x k 1,2 = (1 α 2 ) β k2w 12 p k + τ 1 k x k 2 = α 2 α k2 W 2 p k + τ k W h = (x 1k 1 + x1k 2 )τ k 1 + p k ω 0k h + = x 1k 2, (x 0k 1 + x0k 2 = x 0k 2, p +1k ω 1k h )τ k +3p 1 =0. (6.1) A naural cidae for a soluion o he firs four equaions of (6.1) is of he form p k =(β) 1 p k, τ k =(β) 1 τ 0k,τ 1 k =(β) 1 τ 1k,whereβ R is he ineres facor, τ 0k R is he presen nominal value of he ax rae on he young τ 1k R is he presen nominal value of he ax rae on he old. 15

17 In his example, consumer 2 is assumed o face a binding credi consrain while he governmen canno personalize his axes. Then i is required ha his saving or borrowing should remain exacly as hey were in he unaxed siuaion. The budge equaion for his consumer when young is (p k + τ 1 k )x0k 2 xm 2 = p k ω0k 2. In he iniial siuaion, wih no axes, his yields a he seady sae which can be rewrien as or as (β 1 p k x 0k 2 ) x m 2 = β 1 ( β 1 p k ω 0k 2 p k (x 0k 2 ω0k 2 )) = xm 2 x m 2 = β 1 x m 2. We should poin ou ha in he absence of axes, a sricly posiive governmen consumpion is financed hrough a permanen defici (see Equaion (SS-C)) implying ha boh aggregae individual savings are non-zero β is differen from uniy. Sinceweassumehaconsumer s2savingsarenoaffeced by he fiscal policy, if b β is he value of he ineres facor obained wih axes, we should have ( β b 1 p k + β b 1 τ 0k )x 0k 2 β b 1 p k ω2 0k = β 1 x m 2 for any. These equaions imply ha b β should remain unaffeced by he inroducion of axes, bβ = β. By replacing he dem funcions, savings are unalered for consumer 2 if 1 β 1 (α 2 α k2 W 2 β 1 p k ω 0k 2 )=α 2 (p k ω2 0k + βpk ω2 1k ) p k ω2 0k = x m 2 (CS) In his example he oal sysem is composed of 11 equaions; 7 normalized prices, 2 normalized incomes, he governmen budge defici equaion Equaion (CS) concerning he individual borrowing/savings consrain. On he oher h, here are 3 prices 16

18 (β is fixed) 8 axes. Hence a soluion o he se of equaions is likely o exis. The following se of values represens he seady-sae equilibrium: p 2 =1.2399, p 3 = , p 4 = ( 0k 1 )4 =(1k 1 )4 =( , , , ). Finally, he saving is now for consumer for consumer 2, so ha aggregae savings are zero. This las resul agrees wih Equaion (SS-C) as in he new siuaion a balanced budge (δ = 0) co-exiss wih β differen from uniy, implying ha he aggregae savings are zero. The previous example illusraes he mechanism for irrelevance. Saring from a seady sae wih non-zero aggregae savings, here is a ax scheme which keeps he ineres rae unchanged while achieving a zero governmen budge defici. In he new siuaion, he governmen is axing he consumers jus enough o balance is budge. Moreover, his ax-ransfer is such ha aggregae, afer ax, savings become zero. In his game, he change in he aggregae savings is compleely done hrough he unresriced consumer. A furher imporan fac is ha he axes required o obain irrelevance are age-anonymous in he sense ha he ax o be applied on he young on he old is he same. The role played by he consumer whose credi consrain is no binding is crucial. Indeed, if all consumers would be kep a heir iniial levels of borrowing or saving, achieving irrelevance would be impossible. This is clearly seen by considering an economy consising of only one consumer. The budge consrain for generaion 1 consumer during his old age can be rearranged as τ 1x 1 = p x 1 + p ω 1 x m 1, while for he consumer in generaion, hisfirs period consrain is Adding he wo, we obain τ x = p x + p ω xm. τ x + τ 1 x 1 = p ω + ω 1 x x 1 x m 1 x m. Considering now he governmen budge defici equaion d = p g τ x + τ 1 x 1 = p g + x + x 1 ω ω 1 + x m 1 + x m = x m 1 + x m, 17

19 so ha he defici is compleely deermined by he aggregae borrowings savings decisions of he consumers. If hese are unaffeced by he fiscal policy, he defici will remain unchanged. The example above shows ha when consumpion ax insrumens are sufficienly diversified, irrelevance of defici resricions may hold. However, in general as was he case wih no credi resricions (see Ghiglino Shell (2000), Example 11), having several ax insrumens is no sufficien for irrelevance, which also depends on he lengh he magniude of he defici resricion. Indeed, even when here are enough insrumens, i is no assured ha qh sk τ h sk is posiive, i.e. we could have for some s (s = 1, 2,... ) some k (k = 1,..., `) hap sk < 0. This would be consisen wih he formal model, bu is, of course, inconsisen wih free disposal of endowmens. As a consequence, he nex proposiion which generalizes he former example, gives only a necessary condiion for irrelevance a sufficien condiion for weak irrelevance. Proposiion (Relevance of defici resricions in economies wih consumpion axes consumer credi resricions): Suppose ha only anonymous consumpion axes are available ha he credi consrain of a leas one consumer is no binding. Le x be an allocaion ha can be implemened as a consiuional equilibrium wih a fiscal policy defici resricion δ le r, 0 r<n,be he number of consumers for which he credi consrain is binding. Then, if n + r>`he defici resricion is weakly ( srongly) relevan. On he oher h, for n + r ` he defici resricion is weakly irrelevan. Proof: When consumers are poenially credi consrained, dem for commodiies may depend on he individual borrowings or lendings, so ha hese mus be kep consan when he policy changes. Formally, x m h,wihxm = p ωh q x h,iskepconsan for consrained consumers. Denoe his quaniy by b h. Furhermore, since here is some consumer whose credi consrain is no binding, prices in successive periods are linked. Therefore, in period he relevan sysem consiss of 2 ` 1 condiions on prices r +n condiions on individual wealhs. Le he consumers whose credi consrain is binding be denoed by h =1,,r while he remaining h = r +1,,n have non-binding credi resricions. Taking ino accoun he resricion on he defici, he sysem of 2 ` + r + n equaions can be wrien as 18

20 bp + bτ = (p 1 + τ 1 )R, p +1 + τ +1 = (p 1 + τ 1 )R +1, p ωh b h = (p 1 + τ 1 )W,h, h =1,,r p ω h + p+1 ω +1 h = (p 1 + τ 1 )W h, h =1,,n P h=n P i=` h=1 i=1 τ i f i h (R,R +1,W h )+τ,i 1f,i 1,h (R 1 1,R 1,W 1,h ) = δ, for i = 1,..., `, where he R s, he W s, he δ s are fixed. Suppose ha n `. Inhe Appendix, i is shown ha i is useful o consider as free variables he las ` n prices of period : p,n+1,...,p ` he firs n prices of period + 1 : p +1,1,...,p +1,n. This sysem, which is linear in 3 ` unknowns, has a soluion if only if n + r `. The usual sign resricions on he p s apply so his condiion is no sufficien for irrelevance. 7. Conclusion We have considered here a general OG exchange economy wih anonymous axes ransfers consrains on individual borrowing. We ask wheher or no he se of equilibrium allocaions is affeced by consiuional resricions on he governmen s budge defici. The credi consrains are imporan. Wih credi consrains on individuals only anonymous lump-sum axes, srong (or global) irrelevance of defici resricions is impossible weak (or local) irrelevance can obain only in unineresing circumsances. This srongly conrass wih he case wihou individual credi consrains, where defici resricions are globally ( weakly) irrelevan. Wih credi consrains on individuals only anonymous consumpion axes, global defici irrelevance is impossible jus as i is for he case wihou credi consrains. If here is a sufficien number of ax insrumens a leas one consumer s credi consrain is no binding, hen here is weak (or local) irrelevance of he defici resricion. This generalizes a similar resul for he model wihou consumer credi consrains. Consumpion axes are beer for avoiding defici resricions han are lump-sum axes. Consumpion axes, even anonymous consumpion axes, provide a mean for ransferring income from one individual o anoher in he same generaion. Wih only anonymous lump-sum axes, inra-generaional ransfers are no possible. The presen paper along wih Ghiglino Shell (2000) indicaes ha here can be limis on he governmen s abiliy o avoid he resricions on is defici. Of course, as 19

21 Kolikoff (1993) ohers have argued, here is sill pleny scope for he governmen o evade (as opposed o avoid ) he defici resricions by alering in is books he iming of receips disbursemens, by guaraneeing off-he-books privae loans, so forh. We sress again ha o say he defici resricion is irrelevan is no o say ha he defici does no maer. I is likely o maer if individuals condiion heir (raional or non-raional) expecaions on he defici. References Balasko, Y., 1988, Foundaions of he heory of general equilibrium, Academic Press, Boson, MA. Balasko, Y., K. Shell, 1980, The Overlapping-Generaions Model, I: he case of pure exchange wihou money, Journal of Economic Theory, 23, Balasko, Y., K. Shell, 1981, The Overlapping-Generaions Model, II: he case of pure exchange wih money, Journal of Economic Theory, 24, Balasko, Y., K. Shell, 1986, Lump sum axes ransfers: public deb in he overlapping generaion model, in Equilibrium analysis: Essays in honor of Kenneh J. Arrow, Volume II, Balasko, Y., K. Shell, 1993, Lump sum axaion: he saic economy, in General Equilibrium, Growh Trade II, Essays in Honor of Lionel McKenzie, Academic Press. Barro, R., Are governmen bonds ne wealh?, 1974, Journal of Poliical Economy, 82, Ghiglino, C. K. Shell, 2000, The Economic Effecs of Resricions on Governmen Budge Deficis, Journal of Economic Theory, 94 (1), Ghiglino, C., M. Tvede, 1995, Endowmens, sabiliy flucuaions in OG models, Journal of Economic Dynamics Conrol, 19, Kolikoff, L. J., 1993, From Defici Delusion o he Fiscal Balance Rule, Journal of Economics, Suppl. 7, Samuelson, P. A., 1958, An Exac Consumpion-Loan Model of Ineres wih or wihou he Social Conrivance of Money, Journal of Poliical Economy, 66, Appendix: Rank compuaions For noaional convenience, we focus aenion on he case n 2. Le he consumers for whom heir credi consrain is binding be h =1,..., r while for he remaining, h = r + 1,...n, heir credi consrain is no binding. Firs, consider a consumer of generaion 0. I is clear from Ghiglino Shell (2000) ha he se of free parameers lef afer 20

22 imposing he condiion of consan individual dems o hese consumers is a se of dimension l n. Le us hen consider as free he las l n prices p 1,n+1,...,p 1,l. Second, consider he consumers of generaion, ( = 1, 2,...), wih he consrain ha he prices p 1,...,p n are already fixed (from previous-period condiions). Using he relaionship beween he prices in periods +1, he sysem of equaions associaed o a given dem can be wrien as P n h=1 P l i=1 τ i f i h (R,R +1 ˆp +ˆτ = (p 1 + τ 1 )R p +1 + τ +1 = (p 1 + τ 1 )R +1 p ωh b h = (p 1 + τ 1 )W p ωh + p+1 ω +1 h = (p 1 + τ 1,W h )+ +τ i 1f i 1,h (R 1,R 1,W 1,h ) = δ h for h =1,...,r )W h for h =1,...,n where he quaniies R R l 1, R 1 R l, Wh, W h δ are fixed. The sysem of 2l 1+n + r +1=2l + n + r equaions becomes linear in 3l unknowns, p,n+1,...,p,l, p +1,1,...,p +1,n, τ τ +1. Inroduce he vecors P0 R l n P1 +1 R n defined by P0 = p,n+1 p,n+2. p,l P +1 1 = p +1,1 p +1,2. p +1,n. In marix form, he sysem can be wrien as A z = b wih 0 0 R I n I l n 0 R 0 I l n I n R I n 0 A = 0 0 R I l n $ 0 W J r ω ω +1 W J n P 0 0 n h=1 f P h 1 n ˆ h=1 fh 0 0 R 2 R = R 3, R. =,R +1 = R +1 = R n n 1 1 R +1,n+1 R +1,n+2. R +1,l l+n 1 R,n+1 R,n+2. R l,$ = l n 1 ω,n+1 1 ω,n+1 2. ω,n+1 r 21 R +1,1 R +1,2. R +1,n ω,n ω l 1 ω,n ω2 l... ω,n+2 r... ωr l n 1 r l n 2l+n+r 3l,,,

23 ω = ω,n+1 1 ω,n+1 2. ω,n+1 n z = P P +1 τ 1 τ 2 ṭ. τ +1,l ω,n ω1 l ω,n ω2 l... n... ω l ω,n+2. n n l n, ω +1 = ω +1,1 1 ω +1,1 2. ω +1,1 n ω +1, ω +1,n 1 ω +1, ω +1,n 2... ω +1,2 n... ω +1,n n Le also W R n W R n be he vecors of individual wealhs. The rank of he marix A is equal o he rank of he marix n n 0 0 R I n 1 0 I l n 0 R 0 I l n $ 0 W J n 0 0 ω ω +1 W J n 0 0 P 0 0 n h=1 f h 1 ˆ P n h=1 fh l+n+r 2l plus l. Some edious manipulaions similar o hose performed in Ghiglino Shell (2000), show ha generically he above marix has maximal rank. Then, for l = n + r he A marix has full rank 3l. Inhiscasehesysemhasalwaysasoluion. Thesame can be said for n + r<l. Suppose now ha l +1=n + r. Then he A marix is a 3l +1 3l marix which has generically maximal rank 3l. Consider he square 3l + 1 marix associaed o he augmened sysem, (A,b )leusproveharank(a,b )=3l + 1. Indeed, he las coordinae of b is a funcion of δ ha can be wrien as δ nx lx h=1 i=1 τ i 1 f i i 1. The deerminan of (A,b )isafirs degree polynomial expression in δ. Therefore, o prove ha he relevan marix has full rank for an open dense se of values of δ i is enough ha he coefficien of δ in he polynomial expression is nonzero, which can be seen o be generically rue. Since Rank(A) <Rank(A, b), he soluion se is empy. This is he borderline case so he same resul holds also whenever n + r>l+1. 22